From elliptic integrals to Diophantine equation
Prof. Dr. Akshay Venkatesh
(Princeton)
When does a polynomial equation in two variables, such as X2-Y5=11, have finitely many solutions in rational numbers? This was a famous question of Mordell, and was settled by Faltings in 1983. Recently Brian Lawrence and I found a new approach to this result. Our proof makes key use of "period mappings," which generalize some aspects of the classical theory of elliptic integrals. I will begin by reviewing some of this theory, starting with ideas contributed by Jacobi and Weierstrass, and then try to give some hint of how one gets from there to number theory.